Nano-indentation of Cubic and Tetragonal Single … of Cubic and Tetragonal Single Crystals by ......

Nano-indentation of Cubic andTetragonal Single Crystals

by

Qin Zhang

Submitted in Partial Fulfillment

of the

Requirements for the Degree

Doctor of Philosophy

Supervised by

Professor John C. Lambropoulos

Department of Mechanical EngineeringThe College

School of Engineering and Applied Sciences

University of RochesterRochester, New York

2008

Curriculum Vitae

The author was born in Nanchang, Jiangxi providence, China on January 26,

1977. She attended high school at Jianggang High School and graduated in 1994.

She enrolled at Hunan University in 1994 and finished her B.S. degree program

in Engineering Mechanics in 1998. After graduation, she was hired by 602 Heli-

copter Research Institute and worked there as a design engineer for one year. She

continued her graduate study at Hunan University and graduated with a Master’s

degree in Engineering Mechanics in 2002.

In Fall 2002, she was accepted into the doctoral program at the University of

Rochester under the supervision of Professor John C. Lambropoulos. She receive

her second Master’s degree in Mechanical Engineering from the University of

Rochester in 2004.

Publications:

• “Residual Stress Model for CaF2.” (with J. C. Lambropoulos) J. Mater.

Res., 22:2796, 2007.

• “A model of CaF2 indentation.” (with J. C. Lambropoulos) J. Appl. Phys.

101 (4), 2007.

• “Approximate bending solutions for thin, shallow shells of variable thick-

ness.” (with J. C. Lambropoulos) SPIE, May, 2005.

• “Comparative microindentation and dislocation activity in Silicon and CaF2:

A model.” (with J. C. Lambropoulos) SPIE, February, 2005.

• “Analysis of thin plates by the local boundary integral equation (LBIE)

method.” (with S. Y. Long) Eng. Anal. Bound. Elem. 26 (8), 2002.

iii

• “Application of the dual reciprocity boundary element method to solution

of nonlinear differential equation.” (with S. Y. Long) Acta. Mech. Solida

Sinica 13 (2), 2000.

Presentations:

• “Indentation of CaF2.” Department of Mechanical Engineering, University

of Rochester, November 10, 2006.

• “Comparative Indentation of Cubic and Tetragonal Crystals.” Department

of Mechanical Engineering, University of Rochester, May 15, 2005.

• “Comparative microindentation and dislocation activity in Silicon and CaF2:

A model.” Poster presentation, SPIE-Optifab, May 5, 2005.

• “Two shell bending problems.” Department of Mechanical Engineering,

University of Rochester, March 3, 2005.

iv

Acknowledgments

I would first like to thank my advisor, Professor John C. Lambropoulos, who

diligently guided me, generously gave of his time, and shared of his extensive

knowledge. I am very grateful for the financial and academic support that he has

provided throughout my graduate studies.

Many thanks to Professor Jame C.M. Li, Professor Stephen J. Burns, Professor

Sheryl M. Gracewski, Professor Renato Perucchio, Professor Paul D. Funkenbusch,

Professor David J. Quesnel, and Professor Hong Yang for their guidance and the

knowledge I have learned from their classes.

I would like to thank Chris Pratt for helping me set up the experiments in my

thesis. Thank you also, to Jill Morris, Carmen Schlenker, and Carla Gottschalk

for all their help along the way.

Finally, I would like to thank my family especially my mother and my husband

for their love and support.

v

Abstract

Due to the nature of their material properties, Calcium Fluoride (CaF2), Magne-

sium Fluoride (MgF2), and Potassium Dihydrogen Phosphate (KDP) are widely

used for industrial purposes. A better understanding of the mechanical properties

of these materials is of technological and scientific importance. The indentation

nano-mechanical response of CaF2 (cubic), MgF2 (tetragonal), and KDP (tetrag-

onal) optical crystals were studied and compared by using nano-indentation tests

and finite element simulation with a mesoplastic formulation. Appropriate val-

ues of material parameters were determined by correlating the load-displacement

curves from numerical simulations with the corresponding experimental data. The

effects of elastic anisotropy and crystallographic symmetry on the load-deflection

curves, surface profiles, contact radius, hardness, stress distributions, and cleavage

underneath the spherical indenter at two stages, namely at maximum indentation

load and after the load has been removed were also examined.

vi

Table of Contents

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 2 Formulation of Crystal Plasticity . . . . . . . . . . . 10

2.1 Meso-plasticity for Single Crystals . . . . . . . . . . . . . 10

2.2 Finite Element Simulation . . . . . . . . . . . . . . . . . 13

2.2.1 Numerical Model for CaF2 . . . . . . . . . . . . 15

2.2.2 Numerical Model for MgF2 . . . . . . . . . . . . 16

2.2.3 Numerical Model for KDP . . . . . . . . . . . . 17

Chapter 3 Load-displacement for CaF2 . . . . . . . . . . . . . . 21

3.1 Nano-indentation Experiments . . . . . . . . . . . . . . . 21

3.1.1 Material . . . . . . . . . . . . . . . . . . . . . . 21

3.1.2 Experiment . . . . . . . . . . . . . . . . . . . . . 21

3.2 Numerical Results for Uniaxial Compression . . . . . . . 23

3.3 Numerical Results for Nano-indentation . . . . . . . . . . 24

3.4 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4.1 Measurable Indentation Parameters . . . . . . . 26

3.4.2 Spherical Hardness . . . . . . . . . . . . . . . . . 27

Chapter 4 Stress and Residual Stress in CaF2 indentation . . 43

4.1 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2 Residual Stress . . . . . . . . . . . . . . . . . . . . . . . 47

4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Chapter 5 Load-displacement for MgF2 . . . . . . . . . . . . . . 60

5.1 nano-indentation Experiments . . . . . . . . . . . . . . . 60

vii

5.1.1 Material . . . . . . . . . . . . . . . . . . . . . . 60

5.1.2 Experiment . . . . . . . . . . . . . . . . . . . . . 60


5.3 Numerical Results for Nano-indentation . . . . . . . . . . 63

5.4 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . 64



Chapter 6 Stress and Residual Stress in MgF2 indentation . . 79

6.1 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.2 Residual Stress . . . . . . . . . . . . . . . . . . . . . . . 81

6.3 Plastic Zone . . . . . . . . . . . . . . . . . . . . . . . . . 82

Chapter 7 Load-displacement for KDP . . . . . . . . . . . . . . 95


7.2 Numerical Results for nano-indentation . . . . . . . . . . 97

7.3 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . 98



Chapter 8 Stress and Residual Stress in KDP indentation . . 108

8.1 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

8.2 Residual Stress . . . . . . . . . . . . . . . . . . . . . . . 110

Chapter 9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Bibliography 122

viii

List of Tables

Table 1.1 Chemical/physical/optical Properties of CaF2 . . . . . . . . . 8

Table 1.2 Chemical/physical/optical Properties of MgF2 and KDP . . . 9

Table 3.1 Numerical radii (µm) of residual projected indent area of (100)/(110)/

(111) planes for CaF2 (a0 and am are indicated in Figure 3.12) . . 42

Table 3.2 Spherical hardness (GPa) of (100)/(110)/(111) planes for CaF2

(H0 and Hm are calculated from a0 and am indicated in Figure 3.12) 42

Table 5.1 Numerical radii (µm) of residual projected indent area of (001)/(101)/

(111) planes for MgF2 (a0 and am are indicated in Figure 3.12 and

“⊥” represents the plane perpendicular to the side II) . . . . . . . 78

Table 5.2 Spherical hardness (GPa) of (001)/(101)/(111) planes for MgF2


Table 7.1 Numerical radii (µm) of residual projected indent area of (001)

and (100) planes for KDP (a0 and am are indicated in Figure 3.12) 107

Table 7.2 Spherical hardness (GPa) of (001) and (100) planes for KDP


ix

List of Figures

Figure 2.1 FEM model. (a) Entire FEM domain for (111) plane indenta-

tion of CaF2 and ABAQUS analysis coordinate system. (b) Detail

of the mesh at the region of contact . . . . . . . . . . . . . . . . . 18

Figure 2.2 Crystallographic planes of CaF2 and ABAQUS analysis coor-

dinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Figure 2.3 Crystallographic planes of MgF2 / KDP and ABAQUS analysis

coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Figure 3.1 Experimental load-displacement curves for (100)/(110)/(111)

plane indentations of CaF2 . . . . . . . . . . . . . . . . . . . . . . 29

Figure 3.2 Numerical stress strain relations of free uniaxial compression

for plane (110) of CaF2 (coordinates x, y, and z are in the crystal-

lographic direction [110], [110], and [001], respectively). Two sets

of material parameters are 80/120 MPa (initial shear strength/self-

hardening coefficient) and 110/100 MPa . . . . . . . . . . . . . . 30

Figure 3.3 Numerical stress strain relationships of constrained uniaxial

compression for plane (110) of CaF2 (coordinates x, y, and z are in

the crystallographic direction [110], [110], and [001], respectively).

Two sets of material parameters are 80/120 MPa (initial shear

strength/self-hardening coefficient) and 110/100 MPa) . . . . . . 31

Figure 3.4 Comparison between numerical and experimental load-displacement

curves for (100) plane indentation of CaF2 (coordinates x and z

are in the crystallographic direction [001] and [010] , respectively).

Three sets of material parameters are 74/180 MPa (initial shear

strength/self-hardening coefficient), 80/120 MPa, and 110/100 MPa 32

x



are in the crystallographic direction [110] and [001] , respectively).


strength/self-hardening coefficient) and 110/100 MPa . . . . . . . 33



are in the crystallographic direction [011] and [211], respectively).


strength/self-hardening coefficient) and 110/100 MPa . . . . . . . 34

Figure 3.7 Elastic/mesoplastic load-deflection relations for (100)/(110)/(111)

plane indentations of CaF2 . . . . . . . . . . . . . . . . . . . . . 35

Figure 3.8 Numerical deformed surfaces at maximum indentation loads

for (100)/(110)/(111) plane of CaF2 . . . . . . . . . . . . . . . . . 36

Figure 3.9 Numerical deformed surfaces after fully unloading for (100)/(110)/

(111) plane of CaF2 . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Figure 3.10 Numerical maximum pile-up heights for (100)/(110)/(111)

plane indentations of CaF2 . . . . . . . . . . . . . . . . . . . . . . 38

Figure 3.11 Numerical contact radii for (100)/(110)/(111) plane indenta-

tions of CaF2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Figure 3.12 Schematic of indentation showing the residual projected indent

radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Figure 3.13 Comparison between numerical and experimental spherical

hardness of CaF2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Figure 4.1 Contour of the radial stresses under maximum indentation load

for (100)/(110)/(111) plane of CaF2 . . . . . . . . . . . . . . . . 52

xi

Figure 4.2 Contour of the axial stresses under maximum indentation load

for (100)/(110)/(111) plane of CaF2 . . . . . . . . . . . . . . . . . 53

Figure 4.3 Contour of the hoop stresses under maximum indentation load

for (100)/(110)/(111) plane of CaF2 . . . . . . . . . . . . . . . . . 54

Figure 4.4 Contour of the residual radial stresses after fully unloading for

(100)/(110)/(111) plane indentation of CaF2 . . . . . . . . . . . . 55

Figure 4.5 Contour of the residual axial stresses after fully unloading for


Figure 4.6 Contour of the residual hoop stresses after fully unloading for


Figure 4.7 Contour of the maximum normal stresses on the cleavage planes

for (100)/(110)/(111) plane indentation of CaF2 . . . . . . . . . . 58

Figure 4.8 Contour of the residual maximum normal stresses on the cleav-

age planes for (100)/(110)/(111) plane indentation of CaF2 . . . . 59

Figure 5.1 Experimental load-displacement curves for (001)/(101)/(111)

plane indentations of MgF2 . . . . . . . . . . . . . . . . . . . . . 67

Figure 5.2 Numerical stress strain relations of free uniaxial compression for

plane (101) of MgF2 (coordinates x, y, and z are in the direction [a 0

c], [c 0 a], and [010], respectively under analysis coordinate system).

Three sets of material parameters are 168/220 MPa (initial shear

strength/self-hardening coefficient), 216/260 MPa, and 250/270 MPa 68

Figure 5.3 Numerical stress strain relations of constrained uniaxial com-

pression for plane (101) of MgF2 (coordinates x, y, and z are in the

direction [a 0 c], [c 0 a], and [010], respectively under analysis coor-

dinate system). Three sets of material parameters are 168/220 MPa

(initial shear strength/self-hardening coefficient), 216/260 MPa,

and 250/270 MPa . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

xii


curves for (001) plane indentation of MgF2 (coordinates x and z are

in the direction [100] and [010], respectively under analysis coordi-

nate system). Three sets of material parameters are 168/220 MPa


and 250/270 MPa . . . . . . . . . . . . . . . . . . . . . . . . . . . 70



in the direction [a 0 c] and [010], respectively under analysis coordi-

nate system). Three sets of material parameters are 168/220 MPa


and 250/270 MPa . . . . . . . . . . . . . . . . . . . . . . . . . . . 71



in the direction [a a 2c] and [110], respectively under analysis coor-

dinate system). Three sets of material parameters are 168/220 MPa


and 250/270 MPa . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Figure 5.7 Elastic/mesoplastic load-deflection relations for (001)/(101)/

(111) plane indentations of MgF2 . . . . . . . . . . . . . . . . . . 73

Figure 5.8 Numerical deformed surfaces at maximum indentation loads for

(001)/(101)/(111) plane of MgF2 (“⊥” represents the plane perpen-

dicular to the side II and side III) . . . . . . . . . . . . . . . . . 74

Figure 5.9 Numerical deformed surfaces after fully unloading for (001)/(101)/

(111) plane of MgF2 (“⊥” represents the plane perpendicular to the

side II and side III . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Figure 5.10 Numerical maximum pile-up heights for (001)/(101)/(111)

plane indentations of MgF2 . . . . . . . . . . . . . . . . . . . . . 76

xiii

Figure 5.11 Numerical contact radii for (001)/(101)/(111) plane indenta-

tions of MgF2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77


for (001)/(101)/(111) plane of MgF2 . . . . . . . . . . . . . . . . 83


for (001)/(101)/(111) plane of MgF2 . . . . . . . . . . . . . . . . 84


for (001)/(101)/(111) plane of MgF2 . . . . . . . . . . . . . . . . 85


(001)/(101)/(111) plane indentation of MgF2 . . . . . . . . . . . . 86


(001)/(101)/(111) plane of MgF2 . . . . . . . . . . . . . . . . . . 87


(001)/(101)/(111) plane of MgF2 . . . . . . . . . . . . . . . . . . 88

Figure 6.7 Contour of the normalized resolved shear stresses with initial

shear strength of slip systems (110)/<001> and (101)/<010> under

maximum load for (001) plane indentation of MgF2 . . . . . . . . 89


shear strength of slip systems (011)/<100> under maximum load

for (001) plane indentation of MgF2 . . . . . . . . . . . . . . . . . 90







xiv







Figure 7.1 Experimental load-displacement curves for (001) and (100)

plane indentations of KDP . . . . . . . . . . . . . . . . . . . . . . 100

Figure 7.2 Numerical stress strain relations of free uniaxial compression

for plane (100) of KDP (coordinates x, y, and z are in the direc-

tion [001], [100], and [010], respectively under analysis coordinate

system). Two sets of material parameters are 265/380 MPa (initial

shear strength/self-hardening coefficient) and 465/600 MPa . . . . 101

Figure 7.3 Numerical stress strain relations of constrained uniaxial com-

pression for plane (100) of KDP (coordinates x, y, and z are in the

direction [001], [100], and [010], respectively under analysis coordi-

nate system). Two sets of material parameters are 265/380 MPa

(initial shear strength/self-hardening coefficient) and 465/600 MPa 102


curves for (001) plane indentation of KDP (coordinates x and z are

in the direction [100] and [010] respectively under analysis coordi-




curves for (100) plane indentation of KDP (coordinates x and z are

in the direction [001] and [010], respectively under analysis coordi-



xv

Figure 7.6 Numerical deformed surfaces at maximum indentation loads

for (001) and (100) plane of KDP . . . . . . . . . . . . . . . . . . 105

Figure 7.7 Numerical deformed surfaces after fully unloading for (001)

and (100) plane of KDP . . . . . . . . . . . . . . . . . . . . . . . 106








(001) and (100) plane of KDP . . . . . . . . . . . . . . . . . . . . 114


(001) and (100) plane of KDP . . . . . . . . . . . . . . . . . . . . 115


(001) and (100) plane of KDP . . . . . . . . . . . . . . . . . . . . 116

1

1 Introduction

Due to the nature of their material properties, Calcium Fluoride (CaF2), Magne-

sium Fluoride (MgF2) and Potassium Dihydrogen Phosphate (KDP) are widely

used for optical instrumental applications. Their chemical, physical and opti-

cal properties have been thoroughly examined [1]. Some of those properties are

displayed in Tables 1.1-1.2.

CaF2 is an insoluble ionic compound that occurs naturally as the mineral flu-

orite. It adopts a cubic structure wherein calcium is coordinated to eight fluoride

anions and each F−1 ion is surrounded by four Ca2+ ions. CaF2 has many im-

portant properties for industrial purposes such as a wide transmission range, low

refractive index, high permeability, and low birefringence. It is widely used in

the laser, infrared (IR), and ultraviolet (UV) optics applications. Some of these

include mirrors, lenses, windows, and prisms. MgF2 occurs naturally as the min-

eral sellaite. It is a tetragonal solid with the rutile structure. MgF2 provides

both a wide transparent range and a high transmissibility and is often used as an

optical window transmitting from the vacuum ultraviolet (VUV) into the IR. It

is also a birefringent material that supplies polarizing optics for the UV region.

KDP is a transparent dielectric material best known for its nonlinear optical and

electro-optical properties. It has been incorporated into various laser systems for

harmonic generation and optoelectrical switching. Above its ferroelectric Curie

2

temperature (123 K), the crystal structure of KDP is tetragonal.

The mechanical properties of CaF2, MgF2, and KDP have also been exam-

ined due to the obvious scientific and technological significance. Evans and Pratt

investigated the structures of dislocations of CaF2 and measured the tempera-

ture dependence of the dislocation density [2]. Boyarskaya et al. studied the

dislocation structures produced by pyramid indentation on the (111) surface of

CaF2 [3]. It was found that the form of the profiles of dislocation rosettes on the

(111) cleavage planes alters with the orientation of these planes relative to the

indentations. Using transmission electron microscopy (TEM), Sherry and Sande

examined the work hardening behavior of CaF2 at 2000oC, 3000oC, and 4000oC

under uniaxial compression [4]. Also, the effect of the temperature of deformation

and the crystallographic orientation of the compression axis on the deformation

behavior were analyzed. Munoz et al. studied the slip systems of CaF2 with

various orientations deformed by compression between 200oC and 6000oC [5]. It

was found that 100 family planes are the easiest to activate and 110 are the

most difficult. Speziale and Duffy measured the second-order elastic constants of

CaF2 under various pressures ranging up to 9 GPa at 200oC [6]. Their calcula-

tions showed the elastic constants increased linearly with the pressure. A recent

review by Ladison et al. has summarized mechanical properties of CaF2 from

microindentation tests, elastic moduli measurement, and cleavage [7].

The optical performance of CaF2 is highly correlated to its surface quality. For

instance, Stenzel et al. investigated laser damage behavior of CaF2 under various

polishing steps [8]. As opposed to conventional hard polish, advanced methods,

such as ductile machining or chemical polishing, lead to a distinct increase in

its damage threshold. Using optical interferometry and atomic force microscopy

(AFM), Retherford et al. examined the effect of surface quality on transmission

performance for the (111) surface of CaF2 [9]. Their results showed that improved

surface quality and lower subsurface damage could lead to a greater increase in

3

transmittance. Kukleva et al. measured the dependence of the coefficient of

specular light reflection on the surface roughness for the (100), (110), and (111)

planes of CaF2 [10]. Their calculations showed the specular reflection coefficient

increased for smoother surfaces.

To thoroughly exploit its optical characteristics, a great deal of effort has

been devoted to investigating the mechanical properties of CaF2 during its sur-

face finishing process to produce high quality finished parts. For example, the

finished surface characteristics and polishing parameters of CaF2 under differ-

ent methods such as magnetorheological finishing (MRF), single-point diamond

turning (SPDT), ultra-precision float polishing, and ultra-precision grinding have

been examined and compared [11; 12; 13; 14]. It was found that microfracturing

and crystallographic anisotropy are the main factors affecting surface preparation.

Structural defects, such as dislocations, are usually generated during material re-

moval. Crack propagation is then initiated at such defects. In addition, the mech-

anisms of microfracturing and material removal are both shown to be dependent

on the crystalline orientation of the work surface. Kukleva et al. measured the mi-

crohardness, grinding hardness, and tensile strength for the (100), (110), and (111)

planes of CaF2 and the effects of the anisotropy of these physical and mechanical

properties of CaF2 on the shape accuracy of a polished surface were investigated

[10]. Yan et al. examined the crystallographic effects of CaF2 in micro/nano-

machining [15]. The finished surface texture and microfracturing mechanism were

found to differ significantly with crystalline orientation. Such research [10; 11;

12; 13; 14; 15] suggests that the crystallographic anisotropy affects the machined

surface roughness and subsurface damage of CaF2 by affecting the degrees of slip

deformation and cleavage fracture.

Compared with CaF2, there is a much smaller body of work concerning the

mechanical behavior of MgF2. Kandil et al. measured the six independent elastic

constants of MgF2 over the temperature range 4.2-300K [16]. Negative tempera-

4

ture dependences were observed for all six constants. Davies measured ultrasoni-

cally the elastic moduli of MgF2 under various pressures ranging up to 7 kbars [17].

Barber examined single crystals of MgF2 by means of chemical etching and op-

tical and electron microscopy [18]. In as-grown crystals, dislocations were found

decorated with impurity particles. Also, low temperature dislocation glide was

observed to proceed by shear in the favor of <100> directions on 110 planes.

Mecholsky et al. analyzed the crack branching patterns in MgF2 disks by using

the fractal geometric approach [19]. Swab and Quinn characterized stable crack

extension of MgF2 around Knoop indentation surface cracks [20]. They suggested

that the crack growth is initiated by indentation-induced residual stresses. Using

high pressure X-ray diffraction, Haines et al. examined the structures and phase

transitions in MgF2 [21]. The transition from the tetragonal rutile-type to an

orthorhombic phase was observed at 9.1 GPa and followed by the transformation

to the cubic phase at near 14 GPa.

As a prominent material in optical applications, it is essential to understand

the mechanical behavior of MgF2 during surface preparation in order to improve its

optical performance. MgF2 is relatively hard and usually shaped with diamond

tools. Reichling et al. investigated ablation thresholds and damage behavior

of MgF2 prepared by diamond turning [22]. They suggested that damage and

ablation are determined by local surface imperfections.

The mechanical properties of KDP have previously been studied by using in-

dentation experiments. Fang and Lambropoulos measured the Vickers and Knoop

micro-hardness of KDP and the resulting cracking on (100) and (001) faces [23].

They reported the anisotropy of the hardness and the crack sizes among these

faces. The fracture toughness was also extracted by assuming elastic and plas-

tic isotropy. Kucheyev et al. studied the deformation behavior of KDP under

spherical nano-indentation [24]. Multiple “pop-in” events were observed during

loading portion in the load-displacement curves. They suggested that slip is the

5

major mode of plastic deformation in KDP and pop-in events are caused by the

initiation of slip. Guin et al. examined the mechanical properties of KDP by the

methods of uniaxial compression, selective etching, and indentation [25]. They

observed plasticity of KDP by indentation at room temperature and identified

two types of slip systems in KDP. The first system consists of slip planes 110,

101, 112, and 123, with a common Burgers vector, <111>/2. The other

slip system was identified as 010 <100>. Shaskolskaya et al. estimated the mi-

crohardness, microbrittleness, and microstrengh of KDP with the aid of Vickers

microindentation [26]. Chen et al. studied the critical condition of brittle-ductile

transition of KDP by carrying out Vickers indentation on the surface (001) with

various loads and various orientation angles [27]. The experimental results by

AFM showed strong anisotropy and suggested that at load small enough, KDP

may only generate ductile deformation (plastic dent). These observations were

then used to analyze the influence factors on the surface quality of crystal KDP

in SPDT.

Indentation experiments are a useful tool for evaluating a variety of mechan-

ical properties of solids. This is especially true for brittle solids. Current tech-

nology allows indentation experiments to be carried out with the load as low as

a few µN . Therefore, it is possible to obtain material characterization at sub-

micrometer scale by nano-indentation. On the other hand, surface preparation,

for example by grinding or polishing, involves a sequence of micro-indentation and

micro-scratching effects [9]. Accordingly, the examination of indentation behav-

ior of CaF2, MgF2, and KDP not only provides a better understanding of their

mechanical properties, but also their performance during surface preparation.

Semi-empirical analyses of indentation mechanics have received extensive at-

tention. Oliver and Pharr established the relationships between material proper-

ties (elastic modulus, hardness, etc.) and their corresponding load-displacement

curves for isotropic materials [28]. Another common technique is to use numerical

6

methods such as FEM and molecular dynamics (MD) simulations to investigate

the indentation related material properties and phenomena (fracture, dislocation

nucleation, microstructure evolution, etc.) [29; 30]. To date, however, a systematic

study of the mechanical properties of CaF2, MgF2, and KDP under indentation,

and specifically nano-indentation has not been reported. The object of this work

is to provide such an investigation using FEM methods and nanoindenation tests.

FEM simulation is often used to supplement experimental methods in the

study of material properties under indentation. The undefined material param-

eters are usually obtained as input parameters in the FEM simulation by fitting

the simulated load-displacement curves to the experimental data. As a result,

the indentation mechanics and material performance can be effectively examined.

With the appropriate material constitutive laws, FEM simulation can model ma-

terial behavior in multiple length scales. There are two general approaches used

to study plasticity: macroplasticity and microplasticity. Macroplasticity is based

on classical continuum mechanics which relies on empirical assumptions for differ-

ent material response. Microplasticity analysis, however, is linked with detailed

material deformation mechanisms and micro-structural parameters. This allows

for microplasticity techniques to distinguish various plastic deformation mecha-

nisms on a microstructural level. Mesoplasticity is introduced as a combination

of solid mechanics and material science. It provides a connection between the

continuum-based macroplasticity and the physical theory of microplasticity.

Plastic deformation of crystalline materials generally takes the form of crys-

talline slip with dislocations gliding along the corresponding slip systems. To

investigate the effects of its crystalline anisotropy and the underlying dislocation

evolution, a mesoplastic formulation with the length scale of crystalline slip is a

suitable approach. A systematic study on the mesoplasticity of single crystals

originates from the historic works of Taylor and Elam [31]. It was here where

plastic slip along various orientations was first observed and examined. A mathe-

7

matical description of the constitutive relations of mesoplasticity for single crystals

is provided by Hill [32] and Hill and Rice [33]. These mesoplastic formulations

can be coded into FEM programs and have been used to solve complex problems.

For instance, Peirce et al. analyzed the nonuniform and localized deformation in

ductile single crystals subject to tensile loading [34], Yoshino et al. investigated

the dislocation generation and propagation during indentation of a single-crystal

silicon [35], Liu et al. examined the mechanical behavior of single crystal cop-

per under spherical indentation [36], and Wang et al. studied the dependence

of nano-indentation pile-up patterns and of microtextures on the crystallographic

orientation of copper single crystals [37].

In this paper, the mechanical properties of CaF2, MgF2, and KDP under spher-

ical indentation were investigated in detail by using nano-indentation tests and

finite element method with a mesoplastic formulation. The mesoplastic constitu-

tive laws were implemented as a user-material subroutine in ABAQUS/Standard

[38]. Indentation on the main crystallographic planes: (100), (110), and (111)

of CaF2; (001), (101), and (111) of MgF2; (100) and (001) of KDP was ana-

lyzed. Appropriate values of material parameters were determined by correlating

the load-displacement curves from numerical simulations with the corresponding

experimental data. We have examined the effects of crystallographic anisotropy

on the load-deflection curves, surface profiles, contact radius, spherical hardness,

stress distributions, and cleavage at two stages, namely at maximum indentation

load and after the load has been removed. Our model results are compared with

available experimental observation of surface microroughness, subsurface damage,

and material removal rate in grinding. This provides a better understanding of

microfractures and crystalline anisotropy of these materials, and their effect on

the surface quality during manufacturing.

8

Table 1.1 Chemical/physical/optical properties of CaF2 [1]

Properties CaF2 (Chemical)

Crystal system/structure Cubic/Fluorite Lattice constant (A0) 5.46

Color Colorless (Physical)

Density 2.329(25°C) Melting point (°C) 1360

Thermal conductivity (cal/cm sec°C) 2.32E-2 (36°C) Thermal expansion (/°C) 24E-6(20 60°C) Specific heat (cal/g°C) 0.204(0°C)

Dielectric constant 6.76 (1MHz Young’s modulus (GPa) 75.8

Shear modulus (GPa) 33.77 Bulk modulus (GPa) 88.41

Rupture Modulus (MPa) 36.5 Hardness (Knoop Number) 160<110>, 178<100>

CRSS (critical shear stress, MPa) 15 Apparent elastic limit (MPa) 36.54

Poisson ratio 0.26 Cleavage plane (111)

Elastic coefficient (GPa) C11/C12/C44 164/53/34

Solubility index (number of grams for 100g of water)

0°C 1.31E-3 20°C 1.51E-3

(Optical) Reflection loss (for 2 surfaces) 5.6% (4µm)

Refractive index 1.39908 (5µm) Transmission range (µm) 0.13 12.0 Reststrahlen Peak (µm) 35

Absorption coefficient (cm-1) dN/dT (/°C) -10.6E-6

9

Table 1.2 Chemical/physical/optical properties of MgF2 and KDP [1]

Properties MgF2 KDP (Chemical)

Crystal system/structure Tetragonal Tetragonal Lattice constant (A0) a=4.64, c=3.06 a=7.453, c=6.975

(Physical) Density (g/cm3) 3.18(25°C) 2.34(25°C)

Melting point (°C) 1255 252 Thermal conductivity (W/m/K) 0.3 at 300K -

Thermal expansion (/K) 13.7E-6(para) 48E-6(perp) -

Specific heat (J/kg/K) 920 Dielectric constant at 1MHz 4.87 (para) 5.45 (perp)

Young’s modulus (GPa) 138.5 33.0 Shear modulus (GPa) 54.66 Bulk modulus (GPa) 101.32 19.8

Rupture Modulus (MPa) Hardness (Knoop Number) 415

Apparent elastic limit (MPa) 49.6 Poisson ratio 0.276 0.3 Cleavage axis C- axis

Elastic coefficient (GPa) C11/C12/C33/C44/C13/C66

140/89/205/57/63/96 72/-6.3/56.4/12.5/15/6.2

Solubility index 0.0002g (100g water) Solubility in acids Soluble

Solubility in organic solvents Unsoluble in alcohol (Optical) Reflection loss (2 surfaces) 5.2% (0.6µm)

Refractive index 1.37608 (0.7µm) 1.494(para) 1.46(perp) at 1.064μm

Transmission range (µm) 0.13-7.0 0.2-1.5 Optical spectral range (mkm) 0.25-1.7

Reststrahlen Peak (µm) 20 Absorption coefficient (cm-1) 0.04 (0.27µm) 0.03

Optical damage threshold (GW/cm2) 5 (λ=1.064μm,

τ =10ns) Nonlinear coefficient (pm/V) D36=0.44

Electro-optical coefficients (pm/V) R41=8.8, R63=10.3

dN/dT (/°C) 2.3 (parallel) 1.7E-6 (perp)

at 0.4µm

dN/dµ=0 1.4µm

10

2 Formulation of Crystal

Plasticity

2.1 Meso-plasticity for single crystals

For crystalline materials, plastic deformation generally takes the form of crys-

talline slip with dislocations gliding along corresponding slip systems. Each slip

system is described by two vectors, the unit normal to the slip plane m and the

unit vector s in the slip plane along the slip direction.

A mathematical description of the constitutive relations of meso-plasticity for

single crystals is provided by Hill, and Hill and Rice based on the pioneering

work of Taylor and Elam [31; 32; 33]. A comprehensive account of reviews and

contributions to this area can be found in Asaro, Peirce, and Needleman [39; 40;

41; 42]. For this reason, we present only a brief overview of the theory.

A mathematical description of kinematics and constitutive relations usually

begins with the computation of the velocity gradient L (= ∂v/∂x) which can be

expressed

L = F · F−1. (2.1)

Here F , the deformation gradient, has the well-known decomposition formula

F = F ∗ · F p (2.2)

11

where F p denotes plastic deformation corresponding to the material flow caused

by plastic slip motion on its active slip systems and F ∗ represents the lattice

distortion and rotation caused by elastic straining and rigid body rotation. It is

assumed that the lattice configuration and the elastic properties are unaffected

by plastic slip. The plastic deformation for a single crystal can be expressed as

F p = 1 +n∑

α=1

γαsαmα (2.3)

where superscript α denotes different slip systems, γα represents the amount of

shear along the slip system (mα, sα), and n denotes the number of active slip

systems. An alternate expression for the second rank tensor L is provided by the

sum

L = D + Ω (2.4)

where D represents the symmetric deformation rate and Ω the skew-symmetric

spin tensor. From equations (2.1) and (2.2), D and Ω can be further decomposed

into their elastic and plastic parts as follows:

D = D∗ + Dp (2.5)

and

Ω = Ω∗ + Ωp. (2.6)

Using equations (2.3) and (2.4), the plastic parts of D and Ω can be related to the

plastic shear rates along corresponding slip systems (denoted γα for α = 1, . . . , n)

by

Dp =n∑

α=1

µα · γα (2.7)

and

Ωp =n∑

α=1

ωα · γα. (2.8)

The dependence of the second rank symmetric tensor µα and skew-symmetric

tensor ωα on the current slip system α of the single crystal is now given by

µα = (sαmα + mαsα)/2 (2.9)

12

and

ωα = (sαmα −mαsα)/2. (2.10)

This, when combined with the above, provides a description of the kinematics

formulation of meso-plasticity for single crystals up to the choice of the current

slip system vectors m and s.

An accessible form of the constitutive relations can be derived from the rate-

form equation for the elastic distortion of crystal lattice

σ∗ = C : D∗. (2.11)

Here C is the fourth rank lattice elasticity tensor and σ∗ is the Jaumann rate of

Kirchhoff stress which rotates with the crystal lattice. Specifically,

σ∗ = σ − Ω∗ · σ + σ · Ω∗ (2.12)

where σ is the ordinary time rate of the Kirchhoff stress. In contrast, the Jaumann

rate of Kirchhoff stress that rotates with the material is given by

σ = σ − Ω · σ + σ · Ω. (2.13)

Combining equations (2.11)-(2.13) and expressing D∗ and Ω∗ in terms of D and

Ω using equations (2.5) and (2.6), it can now be obtained that

σ = C :

D −n∑

α=1

µα + C−1 : (ωα · σ − σ · ωα)

γα

. (2.14)

From this, the constitutive relations of single crystals are completely defined pro-

vided that the plastic shear rates γα, for α = 1, . . . , n, are known. The choice

of the plastic shear rates can lead to either a rate-independent or rate-dependent

formulation.

13

2.2 Finite element simulation

Three dimensional FEM indentation models for the corresponding crystallographic

planes of CaF2, MgF2, and KDP were created using ABAQUS/Standard 6.3. For

a conical indenter with a flank angle of 90 degrees and 10 µm tip radius which was

later used for CaF2 and MgF2 indentation experiments, the material essentially

contacts with its spherical tip up to the depth of 2 µm. This depth is much above

the maximum indentation depth under the various external loads for both materi-

als. Thus, for simplicity, a spherical indenter with radius of 10 µm was used in the

simulation for these two materials. For KDP, a spherical indenter with radius of

1 µm was used in the simulation to match the indentation test done by Kucheyev

et al.[24]. A major concern for the simulation accuracy and efficiency is modeling

the semi-infinite body. Due to the crystalline symmetry, planes (100)/(110)/(111)

of CaF2 have four-, two-, and three-fold rotational symmetry, respectively. Be-

cause of the reflection symmetry within each rotational section, one-eighth (or 45

degrees) of the indented body, which was modeled as cylinder, was numerically

analyzed for (100) plane indentation, and one-fourth (90 degrees) and one-sixth

(60 degrees) of the indented body were used for the (110) and (111) planes in-

dentation, respectively. On the other hand, planes (100)/(001)/(101)/(111) of

MgF2 and KDP have two-, four-, two-, and zero-fold rotational symmetry, respec-

tively. Therefore, one-fourth (90 degrees), one-eighth (45 degrees), one-fourth,

and half (180 degrees) of the indented body were used for the (100), (001), (101)

and (111) planes indentation, respectively. As an illustration, the entire domain

of (111) plane indentation of CaF2 and the ABAQUS analysis coordinate system

are displayed in Figure 2.1(a). For all three plane indentations, the radius of the

indented body is 100 micron and the height is 50 micron. The indented body

is bounded by five characteristic surfaces, labeled as surfaces I-V. Surface I is

the indented surface. The surfaces II-IV can only displace in their own planes.

The cylindrical surface V is traction free. The indentation simulations were made

14

along the y-direction. For CaF2 with a cubic structure, this is also the [100], [110],

and [111] direction for the (100), (110), and (111) crystallographic plane inden-

tation, respectively (Figure 2.2). Whereas for MgF2 and KDP with a tetragonal

structure whose crystallographic direction is not perpendicular to a plane having

the same indices, this direction is the [100], [001], [c 0 a], and [c c a] direction for

the (100), (001), (101), and (111) crystallographic plane indentation, respectively

(Figure 2.3). Figure 2.1(b) shows the detailed mesh at the region of contact (x−y

plane). Approaching the region of contact, a refined mesh was generated in order

to obtain the convergent contact solution.

Three dimensional isoparametric linear brick elements were used to discretize

the half-space. During the simulation, 2720 elements, 3536 elements, 5440 ele-

ments, and 10880 elements were used in the CaF2 (100)/MgF2 (001)/KDP (001),

CaF2 (111), CaF2 (110)/MgF2 (101)/KDP (100), and MgF2 (111) plane indenta-

tion, respectively. The size of the smallest elements is 0.125 micron. Four FEM

models for the corresponding crystallographic planes with this mesh density were

verified to be sufficiently precise to represent the semi-infinite body and to con-

verge to the right solutions. This was done by comparing the numerical results

of the elastic pressure distribution underneath the spherical indenter with Willis’

analytical solution [43]. The spherical indenter was described by an analytical

rigid surface with infinite modulus as provided by ABAQUS. Therefore no dis-

cretization was necessary for the rigid indenter. The contact between the indenter

and material was modeled as frictionless. In this calculation, the vertical displace-

ment was applied to the reference node of the analytical rigid surface until the

maximum indentation depth was attained. The indentation force can be obtained

by calculating the reaction force at the reference node.

15

2.2.1 Numerical Model for CaF2

CaF2 has a cubic structure with three main orientations: (100), (110), and (111).

This leads to its anisotropic material properties. Three elastic stiffness constants

C11, C12, and C44 are needed to define the material behavior. These are taken to

be 168.16 GPa, 48.54 GPa and 33.81 GPa, respectively [6; 44]. Unlike isotropic

materials, the Youngs modulus E is dependent on direction. Its values along

<100>, <110>, and <111> directions are 168.2 GPa, 142.2 GPa, and 133.5 GPa,

respectively, and the anisotropic constant A = 2C44/(C11 − C12) = 0.56 (it is 1.0

for isotropic solids). This shows that for CaF2, <100> direction is the stiffest,

while <111> direction is the most compliant.

The elastic constants C11, C12, and C44 were used to analyze the (100) plane

indentation. The study of the (110) and (111) planes necessitates an appropriate

coordinate transformation

C ′ijk` = aim · ajn · ako · a`p · Cmnop (2.15)

where C ′ijk` and Cmnop correspond to the elastic stiffness matrix (fourth rank

elastic tensor in equation (2.11)) and the aij represent the direction cosines of

axes (a11 = ox′1 · ox1, a12 = ox′

1 · ox2, etc.). The tensor symmetry allows the

transformation process to be reduced to twenty-one independent transformation

equations, each composed of twenty-one terms.

At room temperature (23oC), CaF2 has six crystallographic slip systems de-

fined by the 100 family of slip planes along the <110> family of slip directions

[5]. This was used to study the (100) plane indentation. The analysis of the (110)

and (111) planes requires a coordinate transformation of the crystallographic slip

directions and the unit normal to each slip plane using

mα′

i = aij ·mαj (2.16)

and

sα′

i = aij · sαj . (2.17)

16

Here the direction cosines aij correspond to equation (2.15).

The shear rates of corresponding slip systems in the constitutive equations

were represented by the rate-dependent power-law relation [45]

γα = γα0

∣∣∣τα/gα∣∣∣(1−µ)/µ(

τα/gα). (2.18)

Here γα denotes a reference shear strain rate and the exponent µ characterize

the rate sensitivity (varying from zero to one). τα and gα represent the Schmid

resolved shear stress and the current shear strength in the slip system α. The

Schmid resolved shear stress is given by the relation

τα = mα · σ · sα (2.19)

where σ is the Kirchhoff stress (equations (3.11)-(3.14)). The evolution of gα is

governed by the formula

gα = gint + hαγα (2.20)

where gint is the initial value of the shear strength and hα is the self-hardening

coefficient. The initial value of the shear strength and the self-hardening coefficient

were assumed to be the same in all slip systems.

2.2.2 Numerical Model for MgF2

As a tetragonal material, MgF2 is anisotropic and six elastic stiffness constants

C11, C12, C13, C33, C44, and C66 are needed to define its elastic behavior. These are

taken to be 140.22 GPa, 89.50 GPa, 62.9 GPa, 204.65 GPa, 56.76 GPa, and 95.7

GPa, respectively[16]. The study of the (001), (101) and (111) planes of MgF2

necessitates an appropriate coordinate transformation for these elastic constants

in the ABAQUS analysis system (Figure 2.1(a)) by using equation (2.15).

At room temperature (23oC), MgF2 has six crystallographic slip systems de-

fined by the 110 family of slip planes along the <001> family of slip directions

17

[16]. Notice that the directions in the crystallographic system might not be the

same as those in the ABAQUS analysis coordinate system due to the tetragonal

structure of MgF2. For example, the crystallographic direction [101] is in fact

the direction [a 0 c] in the analysis coordinate system. The study of the (001),

(101), and (111) planes of MgF2 requires a coordinate transformation of the slip

directions and the unit normal to each slip plane in the analysis system using

equations (2.16) and (2.17).

The shear rates of corresponding slip systems in the constitutive equations of

MgF2 were taken to be the same as equations (2.18)-(2.20) with the assumptions

that the initial value of the shear strength and the self-hardening coefficient were

the same for all slip systems.

2.2.3 Numerical Model for KDP

Similar as MgF2, KDP has a tetragonal crystal structure. The six elastic stiffness

constants C11, C12, C13, C33, C44, and C66 for KDP are taken to be 71.65 GPa,

-6.27 GPa, 14.94 GPa, 56.40 GPa, 12.48 GPa, and 6.21 GPa, respectively[25]. At

room temperature (23oC), two types of slip systems were identified in KDP. The

first system consists of slip planes 110, 101, 112, and 123, with a common

Burgers vector, <111>/2. The other slip system was identified as 010 <100>

[25]. The study of the (100) and (001) planes of KDP necessitates an appropriate

coordinate transformation for the elastic constants, the slip directions and the

unit normal to each slip plane in the ABAQUS analysis system using equations

(2.15)-(2.17).

The shear rates of slip systems in the constitutive equations and the assump-

tions about the initial shear strength and the self-hardening coefficient follow the

analysis of CaF2 and MgF2 in the previous sections.

18

(a)

(b)

Figure 2.1 FEM model. (a) Entire FEM domain for (111) plane indentation of CaF2 and ABAQUS analysis coordinate system. (b) Detail of the mesh at the region of contact.

19

Figure 2.2 Crystallographic planes of CaF2 and ABAQUS analysis coordinate system.

20

Figure 2.3 Crystallographic planes of MgF2 / KDP and ABAQUS analysis coordinate system.

21

3 Load-displacement for CaF2

3.1 Nano-indentation Experiments

3.1.1 Material

Three oriented crystallographic faces of CaF2 (100), (110), and (111) were chosen

for the nano-indentation experiments. These faces were grown and cut to specific

orientations (ISP Optics, Irvington, N.Y.). Following polishing to optical stan-

dards, the samples were examined in a Zygo NewView 5000 white light interfer-

ometer (Zygo Corp., Middlefield, C.T.). The RMS (measured surface roughness)

averaged among 3 measures of (100)/(110)/(111) faces are 0.18 nm, 0.35 nm, and

0.26 nm, respectively. Such values of surface roughness are representative of high

precision optical-quality polishing.

3.1.2 Experiment

nano-indentation experiments were performed on three crystallographic planes

(100)/(110)/(111) of CaF2. These experiments were carried out using a nano-

indentation Instruments II (Nano Instruments, Inc., Oak Ridge, Tennessee). This

instrument has a displacement resolution of 0.04 nm and load resolution of 50

22

nN . The experiments were performed at room temperature (23oC). The indenter

is conical with a 90 degrees included angle blending tangentially with a spherical

tip of 10 µm radius. Load-deflection curves with maximum loads of 5 mN , 10

mN , and 15 mN obtained from experiments are shown in Figure 3.1. For the

trial with a maximum load of 10 mN , the case we simulated using finite elements,

the indentation depths of (100)/ (110)/ (111) crystallographic planes are 96 nm,

102 nm, and 126 nm, respectively.

23

3.2 Numerical Results for Uniaxial Compression

To facilitate the numerical simulation of the constitutive laws for the three crys-

tallographic planes of CaF2, the explicit relations derived in the previous chapter

were coded into the user material subroutine UMAT provided by ABAQUS. Two

simple cases, free uniaxial compression and constrained uniaxial compression, were

used to test the user material subroutine. Both cases were simulated using a single

8-node 3D solid element. The material was first compressed to a maximum strain

εmaxyy = −0.05 and then pulled back to zero strain. The elastic stiffness constants

and slip systems were chosen to correspond to the (110) crystallographic plane of

CaF2 under ABAQUS analysis coordinate system (Figure 2.1(a)). In the simu-

lation, the reference shear strain rate γα0 was taken to be 0.001 s−1 and the rate

sensitivity exponent µ was taken to be 0.05. The choice of the values of these

two constants are commonly used for the mesoplasticity formulation (see, for in-

stance, [35; 36; 37]). Two sets of the material parameters, 80/120 MPa (initial

shear strength/self-hardening coefficient) and 110/100 MPa, were used for com-

parison. The choice of these material parameters will be further discussed in the

following section, where we will demonstrate that such parameters indeed describe

the mesoplastic deformation of CaF2.

The stress strain relationships from the numerical simulations are shown in

Figures 3.2 and 3.3. For both cases, the relations between axial stress σyy and axial

strain εyy are displayed. It can be observed that the material behaves harder for

the case of constrained compression due to the surrounding material constraining

the deformation. Since the material is anisotropic, the transverse terms of stress

and strain (if they exist) will be different. For simplicity, in the case of free

compression (σxx = σzz = 0), only the transverse strain εxx versus the axial strain

is displayed. Similarly, for constrained compression (εxx = εzz = 0), only the

transverse stress σxx versus the axial strain is shown. Mathematica computations

24

were used to verify that the FEM results were in strong agreement with easily

derived analytical solutions.

3.3 Numerical Results for Nano-indentation

Before applying the user material subroutine to the nano-indentation problems,

the simulation of (100) plane indentation using one-eighth (45 degrees) of the

indented body and full (360 degrees) indented body were compared to verify

that the FEM models can capture the material symmetries. To investigate the

sensitivity of mesh size on the convergence of the numerical solution, the three

additional mesh lengths of 0.1 micron, 0.15 micron, and 0.25 micron were tested

for the model of (100) plane indentation and compared to the result of the mesh

length of 0.125 micron. The deviations of the load-displacement curves from the

0.1 micron and 0.15 micron cases were within 5% of the curve obtained from the

0.125 micron case. For the 0.25 micron mesh size, the deviation of the results was

11%.

The combined FEM-nano-indentation approach was then used to determine

the material properties for CaF2. Appropriate values of the initial shear strength

and self-hardening coefficients hα in the mesoplastic constitutive relations for the

corresponding crystallographic planes of CaF2 were obtained after multiple sim-

ulations by correlating the FEM results with the experimental loading-deflection

curves. The experimental tests and the finite element results for nano-indentation

with a maximum load of 10 mN are shown in Figures 3.4-3.6. It can be seen

that parameters of 74/180 MPa (initial shear strength/self-hardening coefficient),

80/120 MPa, and 110/100 MPa provide a reasonable numerical approximation to

the experimental tests for the (100), (110), and (111) plane indentations, respec-

tively. For a further comparison, the numerical results of plane (100) indentation

with parameters of 80/120 MPa and 110/100 MPa, the plane (110) indentation

25

with parameters of 110/100 MPa, and the plane (111) indentation with parameters

of 80/120 MPa are also displayed.

Here we will also observe that the high optical quality of the CaF2 surfaces used

(surface roughness in range 0.18-0.35 nm RMS) imply minimal, if any, subsurface

damage induced by the polishing process. Subsurface damage is estimated to be

less than (2-5) × the surface microroughness, i.e., less that 0.4-2.0 nm. On the

other hand, the penetrations used in the nano-indentation experiments are of order

50-150 nm. We have assumed, therefore, that any initial surface damage induced

by the polishing process cannot significantly alter the measured nano-indentation

results.

26

3.4 Data Analysis

3.4.1 Measurable Indentation Parameters

Once the finite element models with the appropriate material parameters for

(100)/(110)/(111) crystallographic planes of CaF2 were obtained, they were then

implemented to examine the nano-indentation behavior of CaF2. Simulations with

maximum loads of 5 mN , 10 mN , and 15 mN were carried out for the (100) and

(111) plane indentations. For (110) plane, indentations with maximum loads of 5

mN and 10 mN were simulated.

The elastic/mesoplastic load-deflection relations for three planes of CaF2 are

shown in Figure 3.7. The result reveals that, for the same indentation load, the

indentation depth of (111) plane is the largest while the depth of the (100) plane is

the smallest. This is partially due to the fact that, for CaF2, the <100> direction

is the stiffest and the <111> direction is the least stiff.

The deformed surfaces at maximum indentation load and after fully unloading

for each of the three planes of CaF2 are displayed in Figures 3.8 and 3.9. It

was observed that for the (100) and (111) plane indentations, the contact profiles

on surfaces II and III are the same. However, for the (110) plane indentation,

this is not the case. This is because the (110) plane has two-fold rotational

symmetry, while the (100) and (111) planes have four- and three-fold rotational

symmetry, respectively. In addition, all three plane indentations exhibit pile-

up after fully unloading, see Figure 3.9. Figure 3.10 shows the maximum pile-

up heights for three planes under different indentation loads obtained from the

deformation curves. It was found that the pile-up for the (100) plane indentation

is slightly higher than that of the (111) plane. For the (110) plane indentation,

pile-up is higher on surface III than that on surface II.

The fact that the contact profiles on surfaces II and III are the same for the

27

(100) and (111) plane indentations indicates that their projected contact areas

are circular. On the other hand, the variation in the contact profiles for the

(110) plane indentation suggests that its projected contact area is elliptical with

semi axes lying in the symmetry planes (surfaces II and III). Figure 3.11 shows

the contact radii of three crystallographic planes under different indentation loads

obtained from the deformation curves. It can be observed that the contact radii of

the (111) plane indentations are larger than those of the (100) plane indentations

and the contact radii on surface III of the (110) plane indentations are larger than

those on surface II.

3.4.2 Spherical Hardness

Hardness represents the resistance of a material to permanent penetration. It

is measured after the indenting force has been removed and some of the elastic

deformation is recovered. For this reason, hardness is important in understanding

a materials finishing process in which abrasive particles are pressed into the sample

surface. Experimentally, hardness is calculated by dividing the maximum external

force by the residual projected contact area. Table 3.1 shows the numerical radii

of the residual projected indent areas for three plane indentations of CaF2, where

the indented radii a0 and am are indicated in Figure 3.12 and were obtained

from the deformation curves after fully unloading. Table 3.2 shows the calculated

spherical hardness of CaF2 at the corresponding maximum indentation loads and

the residual projected indent areas. Notice that for (110) plane indentation, the

residual projected indent area is elliptical. Because of the axisymmetric nature

of the spherical indenter, the anisotropy of the hardness is merely due to the

material anisotropy. The numerical spherical hardness of CaF2 is compared with

the experimental spherical hardness results of Ladison et al. [7] in Figure 3.13.

The individual hardness value was found at the associated d/D, which represents

the ratio of the indent diameter (residual projected indent diameter for numerical

28

results) to the spherical indenter diameter. Their data are within the range of the

results in this work.

29

Figure 3.1 Experimental load-displacement curves for (100)/(110)/(111) plane indentations of CaF2.

30

Figure 3.2. Numerical stress strain relations of free uniaxial compression for plane (110) of CaF2 (coordinates x, y, and z are in the crystallographic direction[1 10], and [ respectively). Two sets of material parameters are 80/120 MPa (initial shear strength/self-hardening coefficient) and 110/100 MPa.

[110], ],001

31

Figure 3.3 Numerical stress strain relationships of constrained uniaxial compression for plane (110) of CaF2 (coordinates x, y, and z are in the crystallographic direction [1 10], and

respectively). Two sets of material parameters are 80/120 MPa (initial shear strength/self-hardening coefficient) and 110/100 MPa.

[110],

],001[

32

Figure 3.4 Comparison between numerical and experimental load-displacement curves for (100) plane indentation of CaF2 (coordinates x and z are in the crystallographic direction

and , respectively). Three sets of material parameters are 74/180 MPa (initial shear strength/self-hardening coefficient), 80/120 MPa, and 110/100 MPa.

]001[ ]010[

33


]011[ and[ , respectively). Two sets of material parameters are 80/120 MPa (initial shear strength/self-hardening coefficient) and 110/100 MPa.

]001

34


]101[ and ]112[ , respectively). Two sets of material parameters are 80/120 MPa (initial shear strength/self-hardening coefficient) and 110/100 MPa.

35

Figure 3.7 Elastic/mesoplastic load-deflection relations for (100)/(110)/(111) plane indentations of CaF2.

36

Figure 3.8 Numerical deformed surfaces at maximum indentation loads for (100)/(110)/(111) plane of CaF2.

37

Figure 3.9 Numerical deformed surfaces after fully unloading for (100)/(110)/(111) plane of CaF2.

38

Figure 3.10 Numerical maximum pile-up heights for (100)/(110)/(111) plane indentations of CaF2.

39

Figure 3.11 Numerical contact radii for (100)/(110)/(111) plane indentations of CaF2.

40

Figure 3.12 Schematic of indentation showing the residual projected indent radius.

41

Figure 3.13 Comparison between numerical and experimental spherical hardness of CaF2.

42

Table 3.1 Numerical radii (μm) of residual projected indent area of (100)/(110)/(111) planes for CaF2 (a0 and am are indicated in Figure 3.12).

CaF2 5mN 10mN 15mN

(100) a0=1.002 a0=1.253 a0=1.503 am=1.252 am= 1.720 am= 2.002

(111) a0= 1.200 a0= 1.600 a0= 1.920 am= 1.506 am= 2.012 am= 2.271

(110) a0 = 1.502(II)/1.004(III) a0= 2.004(II)/1.257(III) am = 1.752(II)/1.253(III) am= 2.254(II)/1.505(III)

Table 3.2 Spherical hardness (GPa) of (100)/(110)/(111) planes for CaF2 (H0 and Hm are calculated from a0 and am indicated in Figure 3.12).

CaF2 5mN 10mN 15mN

(100) H0= 1.585 H0= 2.027 H0= 2.114

Hm= 1.015 Hm= 1.076 Hm= 1.191

(111) H0= 1.105 H0= 1.243 H0= 1.295

Hm= 0.702 Hm= 0.786 Hm= 0.926

(110) H0= 1.055 H0= 1.264

Hm= 0.725 Hm= 0.938

42

43

4 Stress and Residual Stress in

CaF2 Indentation

In order to optimize the surface quality (i.e. minimize surface roughness and

subsurface damage) in manufactured high precision CaF2 surfaces, one must un-

derstand the interaction between abrasive grains used in polishing or grinding with

the CaF2 surface. The interaction is described in terms of the stresses induced

on the CaF2 surface during the “loading” portion of the abrasive grain-surface

interaction, as well as during the “unloading” portion, i.e. the residual stresses

remaining at the surface. Indentation testing is a practical means to provide

fundamental information on damage modes for brittle solids in manufacturing.

During a complete cycle of loading and unloading, the stress redistribution under

the indenter tip may lead to the formation of different types of cracks. Experimen-

tal observations show three potential crack patterns, namely radial, median, and

lateral, for crystalline materials during elastic-plastic indentation [46]. The radial

crack is generated parallel to the load axis, and remains close to the surface. It

is often used in the measurement of fracture toughness for brittle materials. The

median and lateral cracks are generated beneath the plastic deformation zone,

propagating parallel and perpendicular to the load axis, respectively. They are

considered to relate with the subsurface damage and material removal rate during

the finishing process [47]. Moreover, for CaF2, cleavage fracture is also an impor-

44

tant concern in its surface preparation, when the normal tensile stresses on the

cleavage planes (111) crystallographic planes exceed a critical value. The exam-

ination of the stress distributions during the indentation loading and unloading

process can provide a better understanding of crack formation, i.e. initiation and

propagation, and its relation to material anisotropy.

45

4.1 Stress

The stress distributions σxx, σyy, and σzz at the maximum indentation loads for

the three plane indentations of CaF2 were obtained from the simulations. Here,

the stress component σxx in the 3D indentation problem is analogous to the radial

stress in the simplified 2D axisymmetric indentation problem. Therefore we refer

to it as the radial stress. Similarly, we refer to the stress terms σzz as hoop stress

and σyy as axial stress. Notice that, due to the characteristics of anisotropy each of

the radial and hoop directions belongs to distinctive crystal orientations. It is clear

that the current coordinate system (xyz) (Figure 2.1(a)) can be used to describe

the stress distributions on surface II. For surface III, a coordinate transformation

is needed to precisely describe the distributions of radial and hoop stresses. The

transformed coordinate system, explicitly (x′y′z′), is rotated counter-clockwise

about the y-axis 45 degrees for plane (100), and 60 and 90 degrees for planes

(110) and (111), respectively.

The stress contours of radial, axial, and hoop stresses at the maximum load

of 10 mN are shown in Figures 4.1-4.3. These contours indicate that, under

maximum applied load, the maximum compressive stresses occur directly beneath

the indenter. The values of the axial compressive stresses were found to be largest

in the (100) indentation and smallest in the (111) indentation. Again, this is

because, for CaF2, the <100> direction is the stiffest which leads to the smallest

contact area and the <111> direction is the least stiff which leads to the largest

contact area. In addition, for (110) indentation, near the indented surface, the

axial compressive stresses were found larger on surface III than those on surface

II. This might cause larger contact radii on surface III compared to those on

surface II. Moreover, tensile stresses were also found near the indented surface or

below the compressive stresses. It can be seen that both radial tensile and hoop

tensile stresses are largest in the (111) indentation and least in the (100) plane

46

indentation, which indicates that the median and radial cracks will most easily be

generated and propagated during the (111) indentation and least likely to occur in

the (100) indentation. The magnitudes of axial tensile stresses for all three plane

indentations are quite small with relatively larger values in the (111) indentation

than those of the other two plane indentations. This suggests that lateral cracks

are less likely to occur during the loading portion and will have better chance to

happen in the (111) indentation. Figure 4.7 shows the maximum normal stresses

among four possible cleavage planes ((111)/(111)/(111)/(111)). Tensile normal

stresses were only observed in the (111) indentation. This implies that, during

loading process, cleavage fracture will only be generated in the (111) indentation.

Additionally, the stress distributions of the (100) and (110) indentations exhibit

much stronger anisotropy than those of the (111) indentation.

47

4.2 Residual Stress

Residual stresses arise from the mismatch between the deformed plastic zone and

the surrounding elastic medium after fully unloading. Residual tensile stresses

reduce the mechanical performance of materials by facilitating crack initiation

and propagation.

The stress contours of residual radial, residual axial, and residual hoop stresses

after fully unloading from the maximum indentation load of 10 mN are presented

in Figures 4.4-4.6. The results show that in the (100) and (110) indentations, both

residual radial tensile and residual hoop tensile stresses are larger than the corre-

sponding radial tensile and hoop tensile stresses at maximum load. Nevertheless,

these two residual tensile stresses in the (111) indentation are slightly smaller than

the corresponding tensile stresses at maximum load. This indicates that, during

the unloading portion, the median and radial cracks will grow in (100) and (110)

indentations, and will be close or stay the same in the (111) indentation. The

residual axial tensile stresses in all three plane indentations are much larger than

the axial tensile stresses at maximum load. Moreover, the residual axial tensile

stresses are largest in the (111) indentation where they occur directly below the

indented area and least in the (100) indentation where they occur under the pe-

riphery of the indented area. This suggests that lateral cracks will grow during

the unloading segment for all three indentations, and the crack size will be largest

in the (111) indentation and least in the (100) indentation. Also, the pattern

of cracking is markedly different between (100) and (111) indentations. Figure

4.8 shows the maximum residual normal stresses among four possible cleavage

planes. The residual normal tensile stresses are more significant than normal ten-

sile stresses at maximum load for all three indentations. In addition, the residual

normal tensile stresses are largest in the (111) indentation and least in the (100)

plane indentation. This indicates that cleavage fracture will be generated during

48

the unloading process for all three plane indentations, and the size of the cleav-

age will be largest in the (111) indentation and smallest in the (100) indentation.

Furthermore, the spatial distribution of the normal tensile stress is markedly dif-

ferent for the three orientations: in (100) indentation, the normal tensile stress is

shallow and localized near the indented area at the surface; in (110) indentation

the normal tensile stress is deeper and still extends to the surface; in (111) inden-

tation, the normal tensile stress is under the surface and very large. Similar to

the stresses distributions at maximum load, the residual stresses distributions of

(100) and (110) indentations exhibit much stronger anisotropy than those of the

(111) indentation.

49

4.3 Conclusion

Microfractures and crystalline anisotropy are two major factors affecting the sur-

face quality of CaF2 during manufacturing, and ultimately affecting its optical

performance. In order to analyze the stresses that are responsible for the dam-

age mode, a finite element analysis of spherical nano-indentation was carried out

that considers elastic-mesoplastic deformation for single crystals. In this analysis,

indentation of the three main crystallographic planes (100)/(110)/(111) of CaF2

was studied and compared to examine the effects of crystalline anisotropy. We

have emphasized stresses at the maximum indenting load and residual stresses

after the load has been removed.

Appropriate material parameters were obtained by correlating the FEM results

and the corresponding experimental load-displacement curves. The simulations

show a value in the range of 74-110 MPa for the initial shear yield strength, and

a value in the range of 100-180 MPa for the self-hardening modulus.

It was found that due to the lower rotational symmetry, the projected con-

tact areas are elliptical instead of circular for the (110) indentation. All three

plane indentations exhibit pile-up after fully unloading. The pile-up for the (100)

indentation is slightly higher than that of the (111) indentation. For the (110)

indentation, pile-up is different along two symmetry planes. The hardness of the

(111) plane was found the lowest among three planes under spherical indentation.

Same observation was also reported for Vickers [10] and Berkovich indentation[48].

Stresses and residual stresses analysis indicate that during the loading cycle,

median and radial cracks are more likely to grow than lateral cracks. All three

crack modes will most easily be generated in the (111) indentation and least

likely to occur in the (100) indentation. This agrees well with the experimental

observations that the subsurface damage is largest in the (111) plane and least in

the (100) plane under the same grinding conditions [48]. During the unloading

50

cycle, lateral cracks tend to grow in all three plane indentations, and will be most

significant in the (111) indentation and least significant in the (100) indentation.

This suggests that the material removal rate is largest in the (111) plane and least

in the (100) plane for the same grinding conditions. This prediction was found in

accordance with the experimental results of Kukleva et al. [10] that the volume

being ground away is largest for (111) surface and least for (100) surface in the

same time interval under identical test conditions.

Since CaF2 cleaves along 111 planes, we have also emphasized the calculation

of the largest tensile stresses across the 111 planes, namely cleavage stresses.

The distributions of the maximum normal stresses and maximum residual nor-

mal stresses among four possible cleavage planes show that during the loading

cycle, cleavage is expected to happen only in the (111) indentation. During the

unloading cycle, all three planes tend to cleave and the cleavage will be most

substantial in the (111) indentation and least in the (100) indentation. This

suggests that (111) indentation is the most brittle and (100) indentation is the

least brittle. Moreover, the spatial distribution of cleavage stresses shows that:

in (100) indentation, the fractures are formed and propagated more readily near

the indented area at the surface; in (110) indentation, the fractures are also more

likely to propagate on the indented surface; in (111) indentation, the fractures

are formed and propagated through the material interior. This is consistent with

the experimental observations that the surface roughness is highest for grinding

the (111) plane and lowest for (100) plane for the same grinding conditions [13;

48].

The predicted spherical indentation hardness also can be compared to the

available experimental data by Ladison et al. [7], see Figure 3.13. Our model pre-

dictions for (100) indentation underestimate the experimentally measured hard-

ness by 10-20%. We will note, however, that the Ladison experimental work went

up to loads of 2 N , whereas the nano-indentation data on which our model was

51

based only reached 15 mN . It is quite possible that at higher loads the more ex-

tensive plastic deformation involves not only constant self-hardening (as we have

assumed in our model) but also cross-hardening that itself may depend on the

amount of plastic strain. Given the relative lack of available experimental data

on hardening in CaF2, especially at temperature close to room temperature, we

have adopted the simplest approach, i.e. allowed only self-hardening at a constant

rate. Such latent hardening has been documented for the case of FCC crystals

such as Al and Cu, and may increase form unity to values as high as 1.6 to 2.2

[49]. Clearly, the possibilities that different slip systems in CaF2 harden at dif-

ferent rates or that hardening of one system affects that of another also must

be examined. We will also note, however, that for finishing operations, such as

polishing, the loads acting on individual abrasives are expected to be small. In

that case, the assumption of constant and uniform hardening for all slip system

seems reasonable, especially given the fact that, at room temperature, CaF2 is

relatively brittle and thus demonstrates a limited amount of ductility.

52

Figure 4.1 Contour of the radial stresses under maximum indentation load for (100)/(110)/ (111) plane of CaF2.

53

Figure 4.2 Contour of the axial stresses under maximum indentation load for (100)/(110)/ (111) plane of CaF2.

54

Figure 4.3 Contour of the hoop stresses under maximum indentation load for (100)/(110)/ (111) plane of CaF2.

55

Figure 4.4 Contour of the residual radial stresses after fully unloading for (100)/(110)/(111) plane indentation of CaF2.

56

Figure 4.5 Contour of the residual axial stresses after fully unloading for (100)/(110)/(111) plane indentation of CaF2.

57

Figure 4.6 Contour of the residual hoop stresses after fully unloading for (100)/(110)/(111) plane indentation of CaF2.

58

Figure 4.7 Contour of the maximum normal stresses on the cleavage planes for (100)/(110)/ (111) plane indentation of CaF2.

59

Figure 4.8 Contour of the residual maximum normal stresses on the cleavage planes for (100)/(110)/(111) plane indentation of CaF2.

60

5 Load-displacement for MgF2

5.1 Nano-indentation Experiments

5.1.1 Material

Three oriented crystallographic faces of MgF2 (001), (101), and (111) were cho-

sen for the nano-indentation experiments. These faces were grown and cut to

specific orientations (ISP Optics, Irvington, N.Y.). Following polishing to opti-

cal standards, the samples were examined in a Zygo NewView 5000 white light

interferometer (Zygo Corp., Middlefield, C.T.). The RMS (root-mean-square) av-

eraged in 3 measurements each of the (001)/(101)/(111) faces are 0.22 nm, 0.30

nm, and 0.21 nm, respectively, i.e. of high optical finished quality.

5.1.2 Experiment

A nano-indentation Instruments II (Nano Instruments, Inc., Oak Ridge, Ten-

nessee) was used for nano-indentation experiments on the three crystallographic

planes (001)/(101)/(111) of MgF2. The experiments were performed at room

temperature (23oC). The conical indenter has a 90 degrees included angle and a

spherical tip of 10 µm radius. Load-deflection curves for the three planes with

61

maximum loads of 5 mN , 10 mN , and 15 mN obtained from experiments are

shown in Figure 5.1. For the trial with a maximum load of 10 mN , the case

we simulated using finite elements, the indentation depths of (001)/(101)/(111)

crystallographic planes are 70 nm, 85 nm, and 80 nm, respectively.

62


The explicit relations of the constitutive laws derived in the chapter 2 for the three

crystallographic planes of MgF2 were programmed into the user material subrou-

tine UMAT provided by ABAQUS. Two simple cases, free uniaxial compression

and constrained uniaxial compression, were used to test the user material sub-

routine. Both cases were simulated using a single 8-node 3D solid element. The

material was first compressed to a maximum strain εmaxyy = −0.05 and then pulled

back to zero strain. The elastic stiffness constants and slip systems were chosen to

correspond to the (101) crystallographic plane of MgF2 under ABAQUS analysis

coordinate system (Figure 2.1(a)). In the simulation, the reference shear strain

rate γα0 was taken to be 0.001 s−1 and the rate sensitivity exponent µ was taken

to be 0.05. Three sets of the material parameters, 168/220 MPa (initial shear

strength/self-hardening coefficient), 216/260 MPa, and 250/270 MPa, were used

for comparison. The choice of these material parameters will be further discussed

in the following section, where we will demonstrate that such parameters indeed

describe the mesoplastic deformation of MgF2.


Figures 5.2 and 5.3. For both cases, the relations between axial stress σyy and

axial strain εyy are displayed. Since the material is anisotropic, the transverse

terms of stress and strain (if they exist) will be different. For simplicity, in the

case of free compression (σxx = σzz = 0), only the transverse strain εxx versus

the axial strain is displayed. Similarly, for constrained compression (εxx = εzz =

0), only the transverse stress σxx versus the axial strain is shown. Mathematica

computations were used to verify that the FEM results were in strong agreement

with easily derived analytical solutions.

63

5.3 Numerical Results for Nano-indentation

The combined nano-indentation-FEM approach was then used to determine the

material properties for MgF2 for a nano-indentation load of 10 mN . In the sim-

ulation, the reference shear strain rate γα0 was taken to be 0.001 s−1 and the

rate sensitivity exponent µ was taken to be 0.05. The experimental tests and the

finite element results for nano-indentation with a maximum load of 10 mN are

shown in Figures 5.4-5.6. It can be seen that parameters of 168/220 MPa (ini-

tial shear strength/self-hardening coefficient), 250/270 MPa, and 216/260 MPa

provide a reasonable numerical approximation to the experimental tests for the

(001), (101), and (111) plane indentations, respectively. For a further comparison,

the numerical results of plane (001) indentation with parameters of 216/260 MPa

and 250/270 MPa, the plane (101) indentation with parameters of 168/220 MPa

and 216/260 MPa, and the plane (111) indentation with parameters of 168/220

MPa and 250/270 MPa are also displayed.

64

5.4 Data Analysis


The finite element models for (001)/(101)/(111) crystallographic planes of MgF2

with the appropriate material parameters were then used to examine the nano-

indentation behavior of MgF2. Simulations with maximum loads of 5 mN , 10 mN ,

and 15 mN were carried out for the (001), (101), and (111) plane indentations.

The elastic/mesoplastic load-deflection relations for three planes of MgF2 are

shown in Figure 5.7. The result reveals that, for the same indentation load, the

indentation depth of (101) plane is the largest while the depth of the (001) plane

is the smallest.


for each of the three planes of MgF2 are displayed in Figures 5.8 and 5.9. It was

observed that for the (001) and (111) plane indentations, the contact profiles on

surfaces II and III are the same. While for the (101) plane indentation, this is

not the case. Moreover, for (111) plane indentation, the contact profiles on the

planes perpendicular to the side II and III are different as those on the side II and

III. This is because the (001) plane has four-fold rotational symmetry, while the

(101) and (111) planes have two- and zero-fold rotational symmetry, respectively.

In addition, all three plane indentations exhibit pile-up after fully unloading, see

Figure 5.9. Figure 5.10 shows the maximum pile-up heights for these three planes

under different indentation loads obtained from the deformation curves. It was

found that the side III and side II of (101) plane indentation has the highest and

lowest pile-up, respectively, among all three plane indentations.

The fact that the contact profiles on surfaces II and III are the same for

the (001) plane indentations indicates that its projected contact area is circular.

On the other hand, the variation in the contact profiles for the (101) and (111)

65

plane indentation suggests that there projected contact area is elliptical with

semi axes lying in the symmetry planes. Figure 5.11 shows the contact radii of

three crystallographic planes under different indentation loads obtained from the

deformation curves. It can be observed that the contact radii of the (001) plane

indentation are smallest among three plane indentations. The contact radii on

surface III of the (101) plane indentation is larger than those on surface II.

66


Table 5.1 shows the numerical radii of the residual projected indent areas for three

plane indentations of MgF2 with maximum loads of 5 mN , 10 mN , and 15 mN

obtained from the deformation curves after fully unloading. Table 5.2 shows the

calculated spherical hardness of MgF2 at the corresponding maximum indenta-

tion loads by dividing the maximum external loads by the corresponding residual

projected contact areas. Notice that for (101) and (111) plane indentation, the

residual projected indent areas are elliptical. Because of the axisymmetric na-

ture of the spherical indenter, the anisotropy of the hardness is merely due to

the material anisotropy. The results show that the hardness of (001) plane has

the largest value among all three plane indentations and the values of the (111)

plane is slightly larger than those of the (101) plane indentation. This explains

the experimental results that at the same indentation load the indentation depth

of the (001) plane is the smallest and the depth of the (101) plane is the largest.

67

Figure 5.1 Experimental load-displacement curves for (001)/(101)/(111) plane indentations of MgF2.

68

Figure 5.2 Numerical stress strain relations of free uniaxial compression for plane (101) of MgF2 (coordinates x, y, and z are in the crystallographic direction c], 0 a[ and[ a], 0 [c 010],

respectively under analysis coordinate system). Three sets of material parameters are 168/220 MPa (initial shear strength/self-hardening coefficient), 216/260 MPa,and 250/270 MPa.

69

Figure 5.3 Numerical stress strain relations of constrained uniaxial compression for plane (101) of MgF2 (coordinates x, y, and z are in the crystallographic direction c], 0 a[ a], 0 [c and [010], respectively under analysis coordinate system). Three sets of material parameters are 168/220 MPa (initial shear strength/self-hardening coefficient), 216/260 MPa,and 250/270 MPa.

70

Figure 5.4 Comparison between numerical and experimental load-displacement curves for (001) plane indentation of MgF2 (coordinates x and z are in the direction[ 100] and[0 respe- 10],

ctively under analysis coordinate system). Three sets of material parameters are 168/220 MPa (initial shear strength/self-hardening coefficient), 216/260 MPa, and 250/270 MPa.

71

Figure 5.5 Comparison between numerical and experimental load-displacement curves for (101) plane indentation of MgF2 (coordinates x and z are in the direction c] 0 a[ and[ , resp- 010]

ectively under analysis coordinate system). Three sets of material parameters are 168/220 MPa (initial shear strength/self-hardening coefficient), 216/260 MPa, and 250/270 MPa.

72

Figure 5.6 Comparison between numerical and experimental load-displacement curves for (111) plane indentation of MgF2 (coordinates x and z are in the direction 2c] a a[ and[ 110], respectively under analysis coordinate system).Three sets of material parameters are 168/220 MPa (initial shear strength/self-hardening coefficient), 216/260 MPa, and 250/270 MPa.

73

Figure 5.7 Elastic/mesoplastic load-deflection relations for (001)/(101)/(111) plane indentations of MgF2.

74

Figure 5.8 Numerical deformed surfaces at maximum indentation loads for (001)/(101)/(111) plane of MgF2 (“⊥ ” represents the plane perpendicular to the side II and side III).

75

Figure 5.9 Numerical deformed surfaces after fully unloading for (001)/(101)/(111) plane of MgF2 (“ ” represents the plane perpendicular to the side II and side III). ⊥

76

Figure 5.10 Numerical maximum pile-up heights for (001)/(101)/(111) plane indentations of MgF2.

77

Figure 5.11 Numerical contact radii for (001)/(101)/(111) plane indentations of MgF2.

78

Table 5.1 Numerical radii (μm) of residual projected indent area of (001)/(101)/(111) planes for MgF2 (a0 and am are indicated in Figure 3.12 and “⊥ ” represents the plane perpendicular to the side II ).

MgF2 5mN 10mN 15mN

(001) a0=0.751 a0=1.005 a0=1.220 am=1.001 am= 1.252 am= 1.503

(101) a0 = 1.502(II)/1.000(III) a0= 2.005(II)/1.256(III) a0= 2.506(II)/1.508(III) am = 1.752(II)/1.252(III) am= 2.504(II)/1.505(III) am= 3.005(II)/1.757(III)

(111) a0 = 1.501(II)/1.003(⊥ ) a0= 2.002(II)/1.256(⊥ ) a0= 2.253(II)/1.509(⊥ ) am = 1.751(II)/1.253(⊥ ) am= 2.502(II)/1.506(⊥ ) a0= 3.003(II)/1.758(⊥ )

Table 5.2 Spherical hardness (GPa) of (100)/(110)/(111) planes for MgF2 (H0 and Hm are calculated from a0 and am indicated in Figure 3.12).

MgF2 5mN 10mN 15mN

(001) H0= 2.822 H0= 3.152 H0= 3.208

Hm= 1.588 Hm= 2.031 Hm= 2.114

(101) H0= 1.056 H0= 1.264 H0= 1.263 Hm= 0.726 Hm= 0.845 Hm= 0.904

(111) H0= 1.057 H0= 1.266 H0= 1.404 Hm= 0.725 Hm= 0.846 Hm= 0.905

79


MgF2 Indentation

6.1 Stress


the three plane indentations of MgF2 were obtained from the simulations. Similar

as before, we refer to the stress terms σxx as radial stress, σzz as hoop stress and

σyy as axial stress. Notice that, due to the characteristics of anisotropy each of

the radial and hoop directions belongs to distinctive crystal orientations. The

current coordinate system (xyz) (Figure 2.1(a)) can be used to describe the stress

distributions on surface II. For surface III, a coordinate transformation is needed to

describe the distributions of radial and hoop stresses. The transformed coordinate

system, explicitly (x′y′z′), is rotated counter-clockwise about the y-axis 45 degrees

for plane (001), and 90 and 180 degrees for planes (101) and (111), respectively.


of 10 mN are shown in Figures 6.1-6.3. These contours indicate that, under

maximum applied load, the maximum compressive stresses occur directly beneath

the indenter. The values of the axial compressive stresses were found to be largest

in the (001) indentation and smallest in the (111) indentation. In addition, for

(101) indentation, near the indented surface, the axial compressive stresses were

80

found larger on surface III than those on surface II. This might cause larger contact

radii on surface III compared to those on surface II. Moreover, tensile axial stresses

were found near the indented surface and tensile radial and hoop stresses were

found below the compressive stresses. It can be seen that radial tensile stresses are

largest in the (101) indentation and least in (001) indentation, while hoop tensile

stresses are largest in the (111) indentation. This indicates that the median cracks

will most easily be generated and propagated during the (101) indentation and

least likely to occur in the (001) indentation, while radial cracks will most easily be

generated during the (111) indentation. The magnitudes of axial tensile stresses

for all three plane indentations are quite small with relatively larger values in the

(101) indentation than those of the other two plane indentations. This suggests

that lateral cracks are less likely to generate during the loading portion and will

have better chance to happen in the (101) indentation.

81

6.2 Residual Stress



in Figures 6.4-6.6. The results shows that in the (001) and (101) indentations,

both residual radial tensile and residual hoop tensile stresses are larger than the

corresponding radial tensile and hoop tensile stresses at maximum load. Never-

theless, these two residual tensile stresses in the (111) indentation are nearly the

same as the corresponding tensile stresses at maximum load. This indicates that,

during the unloading portion, the median and radial cracks will grow in (001) and

(101) indentations, and will be close or stay the same in the (111) indentation.

The residual axial tensile stresses in all three plane indentations are much larger

than the axial tensile stresses at maximum load. Moreover, the residual axial ten-

sile stresses are largest in the (111) indentation where they occur directly below

the indented area and smaller in the (001) indentation where they occur under

the periphery of the indented area. For (101) indentation, the residual tensile

stresses are only observed on surface III under the periphery of the indented area.

This suggests that lateral cracks will grow during the unloading segment for all

three indentations, and the crack size will be largest in the (111) indentation and

least in the (101) indentation. Also, the pattern of cracking is markedly different

between (101) and (111) indentations.

82

6.3 Plastic Zone

The plastic zones for (001)/(101)/(111) plane indentations of MgF2 were also ex-

amined. This is performed by comparing the resolved shear stresses with the

initial shear strength for the corresponding six slip systems 110/<001> at the

maximum indentation load. The initial shear strength is assumed to be the same

for all slip systems and has the value of 168 MPa, 250 MPa, and 216 MPa for the

(001), (101), and (111) plane indentation, respectively obtained from the simu-

lation results. The resolved shear stresses were calculated using equation (2.19).

Figures 6.7-6.12 show the contour of the normalized resolved shear stresses with

the initial shear strength of individual slip systems for the three plane indentations

of MgF2. Five out of six slip systems for all three plane indentations are found

have positive normalized values. In the simulation, crystalline slip happens where

these normalized values are larger than one and these areas are considered to have

plastic deformation. The results show that for (001) plane indentation, the slip

systems (110)/[001] and (011)/[100] are the major contribution to the plastic de-

formation, and for (101) plane indentation, these slip systems are (110)/[001] and

(101)/[101]. For (111) plane indentation, all six slip systems have decent amount

of crystalline slip. The results also suggest that the size of the plastic zone is

largest for (001) plane indentation and least for (101) plane indentation.

83

Figure 6.1 Contour of the radial stresses under maximum indentation load for (001)/(101)/ (111) plane of MgF2.

84

Figure 6.2 Contour of the axial stresses under maximum indentation load for (001)/(101)/ (111) plane of MgF2.

85

Figure 6.3 Contour of the hoop stresses under maximum indentation load for (001)/(101)/ (111) plane of MgF2.

86

Figure 6.4 Contour of the residual radial stresses after fully unloading for (001)/(101)/(111) plane indentation of MgF2.

87

Figure 6.5 Contour of the residual axial stresses after fully unloading for (001)/(101)/(111) plane indentation of MgF2.

88

Figure 6.6 Contour of the residual hoop stresses after fully unloading for (001)/(101)/(111) plane indentation of MgF2.

89

Figure 6.7 Contour of the normalized resolved shear stresses with initial shear strength of slip systems (110)/<001> and (101)/<010> under maximum load for (001) plane indentation of MgF2.

90

Figure 6.8 Contour of the normalized resolved shear stresses with initial shear strength of slip systems (011)/<100> under maximum load for (001) plane indentation of MgF2.

91


92


93


94


95

7 Load-displacement for KDP

nano-indentation experiments on the two crystallographic planes (001) and (100)

of KDP were performed by Kucheyev et al. with a Hysitron Tribo-Scope nanoin-

dentaion system [24]. The tests were performed at room temperature (23oC) with

an 1 micron radius spherical indenter and under maximum load of 1 mN . The

corresponding load-deflection curves obtained from these experiments are shown

in Figure 7.1. It shows that, the indentation depths of (001) and (100) crystallo-

graphic planes are 90 nm and 112 nm, respectively.


The explicit relations of the constitutive laws derived in the chapter 2 for the two

crystallographic planes of KDP were programmed into the user material subrou-

tine UMAT provided by ABAQUS. Two simple cases, free uniaxial compression

and constrained uniaxial compression, were used to test the user material sub-

routine. Both cases were simulated using a single 8-node 3D solid element. The

material was first compressed to a maximum strain εmaxyy = −0.05 and then pulled

back to zero strain. The elastic stiffness constants and slip systems were cho-

sen to correspond to the (100) crystallographic plane of KDP under ABAQUS

analysis coordinate system (Figure 2.1(a)). In the simulation, the reference shear

96

strain rate γα0 was taken to be 0.001 s−1 and the rate sensitivity exponent µ was

taken to be 0.05. Two sets of the material parameters, 265/380 MPa (initial shear

strength/self-hardening coefficient), and 450/600 MPa, were used for comparison.

The choice of these material parameters will be further discussed in the follow-

ing section, where we will demonstrate that such parameters indeed describe the

mesoplastic deformation of KDP.


Figures 7.2 and 7.3. For both cases, the relations between axial stress σyy and

axial strain εyy are displayed. Since the material is anisotropic, the transverse

terms of stress and strain (if they exist) will be different. For simplicity, in the

case of free compression (σxx = σzz = 0), only the transverse strain εxx versus

the axial strain is displayed. Similarly, for constrained compression (εxx = εzz =

0), only the transverse stress σxx versus the axial strain is shown. Mathematica

computations were used to verify that the FEM results were in strong agreement

with easily derived analytical solutions.

97

7.2 Numerical Results for nano-indentation

The combined nano-indentation-FEM approach was then used to determine the

material properties for KDP for a nano-indentation load of 1mN . In the sim-

ulation, the reference shear strain rate γα0 was taken to be 0.001 s−1 and the

rate sensitivity exponent µ was taken to be 0.05. The experimental tests and

the finite element results for nano-indentation with a maximum load of 1mN are

shown in Figures 7.4 and 7.5. It can be seen that parameters of 450/600 MPa

(initial shear strength/self-hardening coefficient) and 265/380 MPa provide a rea-

sonable numerical approximation to the experimental tests for the (001) and (100)

plane indentations, respectively. For a further comparison, the numerical results

of plane (001) indentation with parameters of 265/380 MPa and the plane (100)

indentation with parameters of 450/600 MPa are also displayed.

98

7.3 Data Analysis


The finite element models for (001) and (100) crystallographic planes of KDP

with the appropriate material parameters were then used to examine the nano-

indentation behavior of KDP.

The load-deflection relations for the two planes of KDP reveal that, for the

same indentation load, the indentation depth of (100) plane is larger than that

of (001) plane. Since the value of the elastic modulus along <100> direction

is higher than that along <001> direction for KDP, the load-deflection curves

suggests that more slip systems were activated during (100) plane indentation

than those during (001) plane indentation. This explains relatively lower initial

shear strength and self-hardening coefficient for (100) plane indentation.


for the two planes of KDP are displayed in Figures 7.6 and 7.7. Due to the

reason that (001) and (100) plane has four- and two-fold rotational symmetry,

respectively, the contact profiles on surfaces II and III are the same for (001) plane

indentation and different for the (100) plane indentation. This also indicates that

the projected contact area is circular for (001) plane indentation and elliptical for

(100) plane indentation with semi axes lying in the symmetry planes. It can be

observed that the contact radius of the (100) plane indentation is very close to

that of the (001) plane indentation and the contact radius on surface III of the

(100) plane indentation is slightly larger than that on surface II. In addition both

two plane indentations exhibit pile-up after fully unloading. Figure 7.7 shows

that the pile-up for the (001) plane indentation is slightly smaller than that of the

(100) plane. For the (100) plane indentation, pile-up is higher on surface II than

that on surface III.

99


Table 7.1 shows the numerical radii of the residual projected indent areas for

two plane indentations of KDP at the maximum load of 1mN obtained from the

deformation curves after fully unloading. Table 7.2 shows the calculated spherical

hardness of KDP by dividing the maximum external loads by the corresponding

residual projected contact areas. Notice that for (100) plane indentation, the

residual projected indent area is elliptical. Because of the axisymmetric nature

of the spherical indenter, the anisotropy of the hardness is merely due to the

material anisotropy. The experimental spherical hardness by Kucheyev et al. [24]

is also included in the Table 7.2 for comparison. Their data are consistent with the

results in this work. Our results show that the hardness of (100) and (001) planes

are very close (∼15% variation) which suggests small hardness anisotropy among

(100) and (001) faces. This was also observed by Fang and Lambropoulos [23].

From our data, it is not obvious which face is relatively harder than the other.

Two opposite conclusions might be drawn based on the choice of the numerical

radii. The measurements of Kucheyev et al. [24] suggest that (001) face is slightly

harder than the (100) face. Since their radius of the residual impression at the

maximum pile up (which is am in the present work) for (001) plane indentation

at 1mN is 0.5 micron (which is 0.505 micron in our results), it appears that they

used a0 for the calculation instead of am as the latter should give a hardness

value of 1.28 GPa and not the 2.0±0.2 GPa. On the other hand, the Vickers

hardness of (100) and (001) faces of KDP measured by Rao and Sirdeshmukh [50]

are approximately 1.43 GPa and 1.30 GPa, respectively, which suggests that (100)

face is slightly harder than the (001) face. Their data matches our results using

the numerical radii of am.

100

Figure 7.1 Experimental load-displacement curves for (001) and (100) plane indentations of KDP. These curves were obtained from the article by Kucheyev et al. [24].

101

Figure 7.2 Numerical stress strain relations of free uniaxial compression for plane (100) of KDP (coordinates x, y, and z are in the direction[ and[ respectively under analysis coordinate system). Two sets of material parameters are 265/380 MPa (initial shear strength/self-hardening coefficient) and 465/600 MPa.

001], [100], 010],

102

Figure 7.3 Numerical stress strain relationships of constrained uniaxial compression for plane (100) of KDP (coordinates x, y, and z are in the direction[ and[ respectively under analysis coordinate system). Two sets of material parameters are 265/380 MPa (initial shear strength/self-hardening coefficient) and 465/600 MPa.

001], [100], 010],

103

Figure 7.4 Comparison between numerical and experimental load-displacement curves for (001) plane indentation of KDP (coordinates x and z are in the direction[ 100] and[0 respec- 10],

tively under analysis coordinate system). Two sets of material parameters are 265/380 MPa (initial shear strength/self-hardening coefficient) and 465/600 MPa. The experimental curve was obtained from the article by Kucheyev et al. [24].

104

Figure 7.5 Comparison between numerical and experimental load-displacement curves for (100) plane indentation of KDP (coordinates x, and z are in the direction[ and[ respe- 001] 010],

ctively under analysis coordinate system). Two sets of material parameters are 265/380 MPa (initial shear strength/self-hardening coefficient) and 465/600 MPa. The experimental curve was obtained from the article by Kucheyev et al. [24].

105

Figure 7.6 Numerical deformed surfaces at maximum indentation loads for (001) and (100) plane of KDP.

106

Figure 7.7 Numerical deformed surfaces after fully unloading for (001) and (100) plane of KDP.

107

Table 7.1 Numerical radii (μm) of residual projected indent area of (001) and (100) planes for KDP (a0 and am are indicated in Figure 3.12).

KDP 1mN

(001) a0=0.375 am=0.505

(100) a0 = 0.460(II)/0.336(III) am = 0.604(II)/0.362(III)

Table 7.2 Spherical hardness (GPa) of (100) and (100) planes for KDP (H0 and Hm are calculated from a0 and am indicated in Figure 3.12).

KDP 1mN 1 mN ( Kucheyev et al. [24] )

(001) H0= 2.26

2.0 ± 0.2 Hm= 1.28

(100) H0= 2.06

1.6 ± 0.2 Hm= 1.46

108


KDP Indentation

8.1 Stress


the two plane indentations of KDP were obtained from the simulations. As before,

we refer to the stress terms σxx as radial stress, σzz as hoop stress and σyy as axial

stress and due to the characteristics of anisotropy each of the radial and hoop

directions belongs to distinctive crystal orientations. Similar as CaF2 and MgF2,

a coordinate transformation is performed to describe the distributions of radial

and hoop stresses for surface III. The transformed coordinate system, explicitly

(x′y′z′), is rotated counter-clockwise about the y-axis 45 degrees for plane (001)

and 90 degrees for plane (100).


of 1 mN are shown in Figures 8.1-8.3. These contours indicate that, under maxi-

mum applied load, the maximum compressive stresses occur directly beneath the

indenter. The values of the axial compressive stresses were found to be larger

in the (001) indentation than those in the (100) indentation. This might due to

the less plastic deformation during the (001) plane indentation. Moreover, ten-

sile stresses were also found at the periphery of the indented area or below the

109

compressive stresses. It was observed that both radial tensile and hoop tensile

stresses for (001) and (100) plane indentations are comparable. This indicates

that during the loading process the median and radial cracks will be generated

and propagated in nearly the same lever for these two plane indentations. The

magnitudes of axial tensile stresses for both plane indentations are quite small

with relatively larger values in the (100) indentation. This suggests that lateral

cracks are less likely to generate during the loading portion and will have better

chance to happen in the (100) indentation.

110

8.2 Residual Stress



in Figures 8.4-8.6. The results show that for (100) indentations, the values of

the residual radial tensile and residual hoop tensile stresses are larger than the

corresponding radial tensile and hoop tensile stresses at maximum load. While for

(001) indentation, these values are nearly stay the same as those at maximum load.

This indicates that, during the unloading portion, the median and radial cracks

will grow in (100) indentations, and will be close or stay the same in the (001)

indentation. This agrees well with the experimental observations that the fracture

toughness of indenting the (001) planes is higher than that of indenting the (100)

planes [23]. The residual axial tensile stresses in these two plane indentations are

much larger than the axial tensile stresses at maximum load. The large residual

axial tensile stresses in both plane indentations occur directly below the indented

area. This suggests that lateral cracks will grow during the unloading segment for

both plane indentations, and the crack size will be larger in the (001) indentation

than that in the (100) indentation. Also, the pattern of cracking is similar for

these two plane indentations.

111

Figure 8.1 Contour of the radial stresses under maximum indentation load for (001) and (100) plane of KDP.

111

112

Figure 8.2 Contour of the axial stresses under maximum indentation load for (001) and (100) plane of KDP.

112

113

Figure 8.3 Contour of the hoop stresses under maximum indentation load for (001) and (100) plane of KDP.

113

114

Figure 8.4 Contour of the residual radial stresses after fully unloading for (001) and (100) plane of KDP.

114

115

Figure 8.5 Contour of the residual axial stresses after fully unloading for (001) and (100) plane of KDP.

115

116

116

Figure 8.6 Contour of the residual hoop stresses after fully unloading for (001) and (100) plane of KDP.

117

9 Summary

Calcium Fluoride (CaF2), Magnesium Fluoride (MgF2) and Potassium Dihydro-

gen Phosphate (KDP) are widely used for optical instrumental applications due

to their important chemical, physical, and optical properties. The optical per-

formance of these materials is highly correlated to their surface quality. For this

reason, it is important to investigate their mechanical properties during the surface

finishing process to produce high quality finished parts. Indentation is a useful

tool for evaluating mechanical properties of solids, especially for brittle solids.

In addition, surface preparation, for example by grinding or polishing, involves a

sequence of micro-indentation and micro-scratching effects [9]. In this thesis, the

indentation behavior of CaF2, MgF2, and KDP was investigated in detail by using

nano-indentation tests and finite element method with a mesoplastic formulation.

Indentation on the main crystallographic planes: (100), (110), and (111) of CaF2;

(001), (101), and (111) of MgF2; (100) and (001) of KDP was analyzed to examine

the effects of crystalline anisotropy.

Appropriate values of material parameters were determined by fitting the nu-

merical load-displacement curves with the corresponding experimental results.

The simulations show that parameters in the range of 74–110 / 100–180 MPa

(initial shear strength / self-hardening modulus), 168–250 / 220–270 MPa, and

265–450 / 380–600 MPa provide a reasonable numerical approximation to the

118

experimental tests for CaF2, MgF2, and KDP, respectively. The finite element

models with appropriate material parameters were then used to examine the in-

dentation performance of these materials.

The predicted spherical indentation hardness is based on the projected circu-

lar area delineated by the pile-up maximum height or the level of zero vertical

displacement. For CaF2, the hardness values are 1.02–2.11, 0.73–1.26, and 0.70–

1.30 GPa for (100), (110) and (111) plane indentation, respectively, within the

load range of 5–10 mN . The experimental spherical hardness for (111) plane of

CaF2 is 1.10–1.40 GPa provided by Ladison et al. [7] at the load of 2 N . The

fact that our model for (111) indentation underestimates the experimentally mea-

sured hardness might be due to the reason that at higher loads the more extensive

plastic deformation involves not only constant self-hardening (as we have assumed

in our model) but also cross hardening that itself may depend on the amount of

plastic strain. For MgF2, the hardness values are 1.59–3.21, 0.73–1.26, and 0.73–

1.40 GPa for (001), (101), and (111) plane indentation, respectively, within the

load range of 5–10 mN . For KDP, the values are 1.28–2.26 and 1.46–2.06 GPa

for (001) and (100) plane indentation, respectively, at the load of 1 mN . The

experimental spherical hardness provided by Kucheyev et al. [24] are 2.0 ± 0.2

and 1.6± 0.2 GPa for (001) and (100) plane indentation, respectively. Their data

are consistent with our results

For CaF2, the indentation depth of (111) plane is the largest and that of (100)

plane is the smallest under the same load. The projected contact areas for (110)

indentation are elliptical instead of circular due to its lower rotational symmetry.

All three plane indentations exhibit pile-up after fully unloading. The pile-up for

the (100) indentation is slightly higher than that of the (111) indentation. For the

(110) indentation, pile-up is different along two symmetry planes. The hardness

of the (111) plane was found the lowest among three planes indentation.

Stresses and residual stresses analysis of CaF2 indicate that during the loading

119

cycle, median and radial cracks are more likely to grow than lateral cracks. All

three crack modes will most easily be generated in the (111) indentation and

least likely to occur in the (100) indentation. This explains the experimental

observations that the subsurface damage is largest in the (111) plane and least in

the (100) plane under the same grinding conditions [48]. During the unloading

cycle, lateral cracks tend to grow in all three plane indentations, and will be most

significant in the (111) indentation and least significant in the (100) indentation.

This is in accordance with the experimental results of Kukleva et al. [10], that

the material removal rate is largest in the (111) plane and least in the (100) plane

for the same grinding conditions.

The distributions of the maximum normal stresses and maximum residual

normal stresses on cleavage planes 111 of CaF2 show that during the loading

cycle, cleavage is expected to happen only in the (111) indentation. During the

unloading cycle, all three planes tend to cleave and the cleavage will be most

substantial in the (111) indentation and least in the (100) indentation. This

suggests that (111) indentation is the most brittle and (100) indentation is the

least brittle. Moreover, the spatial distribution of cleavage stresses shows that:

in (100) indentation, the fractures are formed and propagated more readily near

the indented area at the surface; in (110) indentation, the fractures are also more

likely to propagate on the indented surface; in (111) indentation, the fractures are

formed and propagated through the material interior. This is consistent with the

experimental observations that the surface roughness is highest for grinding (111)

plane and lowest for (100) plane for the same grinding conditions [13; 48].

For MgF2, the indentation depth of (101) plane is the largest while the depth

of (001) plane is the smallest at the same load. The projected contact areas for

(101) and (111) indentation are elliptical due to their lower rotational symmetry.

All three plane indentations exhibit pile-up after fully unloading. The hardness

of (001) plane has the largest value among all three plane indentations and the

120

value of the (111) plane is slightly larger than that of the (101) plane indentation.

Stresses and residual stresses analysis of MgF2 indicate that during the load-

ing cycle, median and radial cracks are more likely to grow than lateral cracks.

The median cracks will most easily be generated during the (101) indentation and

least likely to occur in the (001) indentation, while radial cracks will most easily

be generated and propagated during the (111) indentation. During the unloading

cycle, lateral cracks will grow during the unloading segment for all three inden-

tations. The lateral crack size will be largest in the (111) indentation where they

occur directly below the indented area and least in the (101) indentation where

they occur under the periphery of the indented area. This suggests that (111)

indentation is the most brittle and (101) indentation is the least brittle.

Plastic zones for (001)/(101)/(111) plane indentations of MgF2 were also ex-

amined by comparing the resolved shear stresses with the initial shear strength

for the corresponding six slip systems 110/<001> at the maximum indentation

load. The results suggest that the size of the plastic zone is the largest for (001)

plane indentation and least for (101) plane indentation.

For KDP, the indentation depth of (100) plane is larger than that of (001)

plane. The projected contact area for (100) indentation is elliptical due to its

lower rotational symmetry. Both plane indentations exhibit pile-up after fully

unloading. The pile-up for (001) plane indentation is slightly smaller than that

of (100) plane. For (100) plane indentation, pile-up is higher on surface II than

that on surface III. The hardness of (100) and (001) planes are very close (∼ 15%

variation) which suggests small hardness anisotropy among (100) and (001) faces.

This was also observed by Fang and Lambropoulos [23].

Stresses and residual stresses analysis of KDP indicate that during the loading

cycle, median and radial cracks are more likely to grow than lateral cracks. The

median and radial cracks will be generated and propagated in nearly the same level

for these two plane indentations. During the unloading cycle, lateral cracks will

121

grow during the unloading segment for both plane indentations. The crack size

will be larger in the (001) indentation than that in the (100) indentation. Also,

the pattern of cracking is similar for these two plane indentations. This suggests

that (001) plane indentation is more brittle than (100) plane indentation.

122

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